Simple Equations – Complete Guide For Class 7 Math Chapter 4

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The chapter on Simple Equations introduces the foundational concepts of algebraic equations, essential for solving mathematical problems involving unknown values. Students learn to form and solve equations using various methods, including trial and error, balancing, and transposing. This chapter emphasizes the importance of understanding variables, writing algebraic expressions, and maintaining equality in equations. Mastery of these concepts is critical for tackling more complex algebraic problems in higher-level mathematics and applying these skills to real-life situations.

Simple Equations

In the chapter Simple Equations, it is made clear that:

• An equation is a mathematical statement that shows the equality of two expressions.
• In algebra, letters (variables) represent unknown numbers. Example: x+7 = 12. Here, x is an unknown number also called ‘variable’.

Variables

Key Points:

• Variable means that can vary. 
• Value is not fixed.
• A variable can take different numerical values.
• Commonly represented by letters such as x, y, z.

What is an Equation?

In Mathematics, an equation is an expression that represents the equality of two expressions containing variables and constants, including an equality sign (=).

The equal sign shows that the value of the expressions to the left of the sign (Left-hand side or LHS) is equal to the value of the expression to the right of the sign (Right-hand side or RHS).

• Example: 2x + 5 = 10 (LHS = RHS).

Now let’s understand how to write an equation

Writing an Algebraic Equation 

First, let’s examine a few equations

1. X+5 > 10 

(This indicates that the value of (x + 5) is greater than 10).

2. X+5 < 10 

(This indicates that the value of (x + 5) is less than 10)

3. X+5 = 10

(This indicates that the value of (x + 5) is equal to 10.)

So, the equation number 3 is the correct way to write an equation.

Key Points for Writing an Equation

Convert statements into algebraic equations. 

Examples:

• “The sum of x and 5 is 10” becomes x + 5 = 10.
• “Thrice a number is 12” becomes 3x = 12.

Property of Equations – An equation remains the same if the L.H.S. and the R.H.S. are interchanged.

Example: 2x+5 = 10 is the same as 10 = 2x+5.

Solutions of Equations: The value of the variable for which the equation is satisfied is called the solution of the equation.

Consider an equation:  5x + 6 = 26.

If x=1, L.H.S. = 5×1+6 =11≠ 26

If x=2, L.H.S. = 5×2+6 = 16 ≠ 26

If x=3, L.H.S. = 5×3+6 = 21 ≠ 26

If x=4, L.H.S. = 5×4+6 = 26 = 26 = R.H.S.

Since the condition is satisfied when x = 4. Hence x = 4 is the solution of the equation 5x + 6 = 26. 

Now let’s understand how to convert the Given Statement into Equation Form

Examples:

• Sentence: The sum of x and 5 is 10. Algebraic Equation: x+5 = 10
• Sentence: Thrice The number is 12. Algebraic Equation: Let x be any number, then 3x = 12
• Sentence: A number divided by 5 is 25. Algebraic Equation: Let the number be x, then x/5 = 25
• Sentence: 6 less than a number is 13. Algebraic Equation: Let the number be x, then x−6 = 13
• Sentence: Ravi has 24 rupees in his piggy bank. How much money does he need to buy a book that costs 50 rupees? Algebraic Equation: Let x be the amount of money Ravi needs. Then x+24 = 50. Subtract 24 from both sides to get x = 26. Ravi needs 26 rupees to buy the book.

Now let’s learn how to convert the given Equation into a Statement form:

Examples:

• Equation:  x−4 = 20. Sentence: Four less than a number is 20.
• Equation: 2x+6 = 24. Sentence: Twice a number increased by six is 24.

Balancing Equations:  A balanced equation is like a weighing balance with equal weights in the two pans: L.H.S. = R.H.S.

Examples:

• Balance the equation: 5 + 3 = 4 + 4 i.e.  L.H.S. = R.H.S.
• Balance the equation: 5 + 3 = 4 + 6 i.e. .L.H.S. ≠ R.H.S

Solving Equations

Two ways of solving the simple equations:

• Balancing the Equation
• Transposing the Equation

Balancing Equation

In the chapter Simple Equations, there are some key points covered about balancing the equations which include:

• If we add the same number to both sides of a balanced equation, the balance is unchanged.
• If we subtract the same number from both sides of a balanced equation, the balance is unchanged.
• If we multiply or divide both sides of the equation by the same number, the balance is unchanged.

Examples:

• Solve the equation: 4x+5 = 21.

                    Step I: Subtract 5 from both sides. 4x+5−5 = 21−5 or 4x=16

                    Step II: Divide by 4 from both sides. 4x/4 = 16/4 = 4

Transposing Numbers

In the chapter Simple Equations, there are some key points covered about transposing numbers which include:

• Changing the side of the number is called transposing a number.
• Transposing a number has the same effect as adding the same number to (or subtracting the same number from) both sides of the equation.
• We can shift a number to the other side of an equation by changing its sign.

The formula for transposing the numbers

Here (m) means the number

• +m goes to the other side as –m. 
• -m goes to the other side as +m.

Example:

• Solve the equation: 4x + 5 = 21 ⟹ 4x + 5 = 21⟹ 4x = 21−5⟹ 4x = 16 ⟹ x = 16/4 = 4 

Applications of Simple Equations to Practical Situations

In the chapter Simple Equations, there are some key points covered about Applications of Simple Equations to Practical Situations. The method is first to form equations corresponding to such situations and then to solve those equations to give the solution to the puzzles/problems.

Example:

• Find a number such that one-fourth of the number is 7 more than 3.

Solution: Let the number be x. One-fourth of x = 4. This number x/4 is more than 3 by 4.

Hence, we get the equation for x as x/4 – 3 = 7. Simplify: x/4 – 3 = 7⟹ x/4 = 7+3 ⟹ x/4 = 10 ⟹ x = 40

• Verification: L.H.S = x/4-3 ⟹ 40/4-3 ⟹ 10-3 = 7 R.H.S.